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Non-Relativistic Propagators Via Schwinger's Method

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Non-Relativistic Propagators Via Schwinger's Method 1260 Brazilian Journal of Physics, vol. 37, no. 4, December, 2007 Non-Relativistic Propagators via Schwinger’s Method A. Aragao,˜ H. Boschi-Filho, C. Farina, Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, 21941-972 Rio de Janeiro, RJ, Brazil. and F. A. Barone Universidade Federal de Itajuba,´ Av. BPS 1303 Caixa Postal 50 - 37500-903, Itajuba,´ MG, Brazil. Received on 10 September, 2007 In order to popularize the so called Schwinger’s method we reconsider the Feynman propagator of two non- relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, R and P, we apply this method by solving the Heisenberg equations for the gauge invariant operators R and p = P ¡ eA, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows explicitly the gauge dependence of the Feynman propagator. Keywords: Schwinger’s method, Feynman Propagator, Magnetic Field, Harmonic Oscillator. I. INTRODUCTION knowledge, since then only a few papers have been written with this method, namely: in 1986, Urrutia and Manterola [18] used it in the problem of an anharmonic charged oscil- In a recent paper [1], three methods were used to com- lator under a magnetic field; in the same year, Horing, Cui, pute the Feynman propagators of a one-dimensional harmonic and Fiorenza [19] applied Schwinger’s method to obtain the oscillator, with the purpose of allowing a student to com- Green function for crossed time-dependent electric and mag- pare the advantadges and disadvantadges of each method. netic fields; the method was later applied in a rederivation The above mentioned methods were the following: the so of the Feynman propagator for a harmonic oscillator with a called Schwinger’s method (SM), the algebraic method and time-dependent frequency [20]; a connection with the mid- the path integral one. Though extremely powerful and ele- point-rule for path integrals involving electromagnetic inter- gant, Schwinger’s method is by far the less popular among actions was discussed in [21]. Finally, pedagogical presen- them. The main purpose of the present paper is to popularize tations of this method can be found in the recent publication Schwinger’s method providing the reader with two examples [1] as well as in Schwinger’s original lecture notes recently slightly more difficult than the harmonic oscillator case and published [22], which includes a discussion of the quantum whose solutions may serve as a preparation for attacking rela- action principle and a derivation of the method to calculate tivistic problems. In some sense, this paper is complementary propagators with some examples. to reference [1]. The method we shall be concerned with was introduced by It is worth mentioning that this same method was inde- Schwinger in 1951 [2] in a paper about QED entitled “Gauge pendently developed by M. Goldberger and M. GellMann in invariance and vacuum polarization”. After introducing the the autumn of 1951 in connection with an unpublished paper proper time representation for computing effetive actions in about density matrix in statistical mechanics [23]. QED, Schwinger was faced with a kind of non-relativistic Our purpose in this paper is to provide the reader with two propagator in one extra dimension. The way he solved this other examples of non-relativistic quantum propagators that problem is what we mean by Schwinger’s method for com- can be computed in a straightforward way by Schwinger’s puting quantum propagators. For relativistic Green functions method, namely: the propagator for a charged particle in a of charged particles under external electromagnetic fields, the uniform magnetic field and this same problem with an addi- main steps of this method are summarized in Itzykson and Zu- tional harmonic oscillator potential. Though these problems ber’s textbook [3] (apart, of course, from Schwinger’s work have already been treated in the context of the quantum action [2]). Since then, this method has been used mainly in rela- principle [18], we decided to reconsider them for the follow- tivistic quantum theory [4–16]. ing reasons: instead of solving the Heisenberg equations for However, as mentioned before, Schwinger’s method is also the position and the canonical momentum operators, R and well suited for computing non-relativistic propagators, though P, as is done in [18], we apply Schwinger’s method by solv- it has rarely been used in this context. As far as we know, this ing the Heisenberg equations for the gauge invariant opera- method was used for the first time in non-relativistic quantum tors R and p = P ¡ eA, the latter being the mechanical mo- mechanics by Urrutia and Hernandez [17]. These authors used mentum operator. This is precisely the procedure followed by Schwinger’s action principle to obtain the Feynman propaga- Schwinger in his seminal paper of gauge invariance and vac- tor for a damped harmonic oscillator with a time-dependent uum polarization [2]. This procedures has some nice proper- frequency under a time-dependent external force. Up to our ties. For instance, we are not obligued to choose a particular A. Aragao˜ et al. 1261 gauge at the beginning of calculations. As a consequence, (i) we solve the Heisenberg equations for X(t) and P(t), and we end up with an expression for the propagator written in write the solution for P(0) only in terms of the operators an arbitrary gauge. As a bonus, the transformation law for X(t) and X(0); the propagator under gauge transformations can be readly ob- tained. (ii) then, we substitute the results obtained in (i) into the ex- In order to prepare the students to attack more complex pression for H (X(0);P(0)) in (3) and using the com- problems, we solve the Heisenberg equations in matrix form, mutator [X(0);X(t)] we rewrite each term of H in a which is well suited for generalizations involving Green func- time ordered form with all operators X(t) to the left tions of relativistic charged particles under the influence of and all operators X(0) to the right; electromagnetic fields (constant Fµn, a plane wave field or (iii) with such an ordered hamiltonian, equation (3) can be even combinations of both). For pedagogical reasons, at the readly cast into the form end of each calculation, we show how to extract the corre- sponding energy spectrum from the Feynman propagator. Al- ¶ i~ K(x;x0;t) = F(x;x0;t)K(x;x0;t); (5) though the way Schwniger’s method must be applied to non- ¶t relativistic problems has already been explained in the litera- with F(x;x0;t) being an ordinary function defined as ture [1, 18, 22], it is not of common knowledge so that we start this paper by summarizing its main steps. The paper is orga- hx;tjH (X(t);X(0))jx0;0i F(x;x0;t) = ord : (6) nized as follows: in the next section we review Schwinger’s hx;tjx0;0i method, in section III we present our examples and section IV is left for the final remarks. Integrating in t, the Feynman propagator takes the form ½ Z ¾ i t K(x;x0;t) = C(x;x0)exp ¡ F(x;x0;t0)dt0 ; (7) II. MAIN STEPS OF SCHWINGER’S METHOD ~ 0 whereRC(x;x ) is an integration constant independent of For simplicity, consider a one-dimensional time- t and t means an indefinite integral; independent Hamiltonian H and the corresponding non- 0 relativistic Feynman propagator defined as (iv) last step is concerned with the evaluation of C(x;x ). This is done after imposing the following conditions h¡iH ti K(x;x0;t) = q(t)hxjexp jx0i; (1) ¶ ~ ¡i~ hx;tjx0;0i = hx;tjP(t)jx0;0i; (8) ¶x 0 ¶ where q(t) is the Heaviside step function and jxi, jx i are the i~ hx;tjx0;0i = hx;tjP(0)jx0;0i; (9) eingenkets of the position operator X (in the Schrodinger¨ pic- ¶x0 0 ture) with eingenvalues x and x , respectively. The extension as well as the initial condition for 3D systems is straightforward and will be done in the next section. For t > 0 we have, from equation (1), that lim K(x;x0;t) = d(x ¡ x0) : (10) t!0+ ¶ h¡iH ti i~ K(x;x0;t) = hxjH exp jx0i: (2) Imposing conditions (8) and (9) means to substitute in their ¶t ~ left hand sides the expression for hx;tjx0;0i given by (7), while Inserting the unity 1l = exp[¡(i=~)H t]exp[(i=~)H t] in the in their right hand sides the operators P(t) and P(0), respec- r.h.s. of the above expression and using the well known re- tively, written in terms of the operators X(t) and X(0) with lation between operators in the Heisenberg and Schrodinger¨ the appropriate time ordering. pictures, we get the equation for the Feynman propagator in the Heisenberg picture, III. EXAMPLES ¶ i~ K(x;x0;t) = hx;tjH (X(0);P(0))jx0;0i; (3) ¶t A. Charged particle in an uniform magnetic field where jx;ti and jx0;0i are the eingenvectors of operators X(t) As our first example, we consider the propagator of a non- and X(0), respectively, with the corresponding eingenvalues relativistic particle with electric charge e and mass m, submit- x and x0: X(t)jx;ti = xjx;ti and X(0)jx0;0i = x0jx0;0i, with ted to a constant and uniform magnetic field B.
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